Introduction

• Signals can be analyzed in both time domain and frequency domain.
• The frequency content of a signal describes how different frequency components contribute to the overall signal.
• Applications in biomedical signals, audio processing, communications, and image processing.

Convolution in Time Domain

• Convolution is a fundamental operation in signal processing.

• Given two signals \(x(t)\) and \(h(t)\), their convolution is defined as:

  \[ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau \]

• In discrete-time, convolution is:

  \[ y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \]

Convolution Theorem

• Convolution in time domain corresponds to multiplication in frequency domain:

  \[ X(f) H(f) = Y(f) \]

• This property is crucial in filter design and system analysis.

Introduction to Fourier Series

(-1.0, 4.0)

(-1.0, 4.0)

Introduction to Fourier Series

• Convolution requiere the representation of the signal in a sum of impulse functions.

• Fourier series represents periodic signals as a sum of sinusoids:

  \[ x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t} \]

  where \(C_n\) are the Fourier coefficients.

• Decomposing a signal into sinusoidal components allows frequency analysis.

Fourier Coefficients

• The Fourier coefficients \(C_n\) are computed as:

  \[ C_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-jn\omega_0 t} dt \]

• Determines how much of each frequency is present in the signal.

Example of Fourier Series Expansion

Example 2 of Fourier Series

(array([-5.,  0.,  5., 10., 15., 20., 25., 30., 35.]), [Text(-5.0, 0, '−5'), Text(0.0, 0, '0'), Text(5.0, 0, '5'), Text(10.0, 0, '10'), Text(15.0, 0, '15'), Text(20.0, 0, '20'), Text(25.0, 0, '25'), Text(30.0, 0, '30'), Text(35.0, 0, '35')])

(array([4.5, 5. , 5.5, 6. , 6.5, 7. , 7.5, 8. , 8.5]), [Text(0, 4.5, '4.5'), Text(0, 5.0, '5.0'), Text(0, 5.5, '5.5'), Text(0, 6.0, '6.0'), Text(0, 6.5, '6.5'), Text(0, 7.0, '7.0'), Text(0, 7.5, '7.5'), Text(0, 8.0, '8.0'), Text(0, 8.5, '8.5')])

Example 2 of Fourier Series

(array([-40., -30., -20., -10.,   0.,  10.,  20.,  30.,  40.]), [Text(-40.0, 0, '−40'), Text(-30.0, 0, '−30'), Text(-20.0, 0, '−20'), Text(-10.0, 0, '−10'), Text(0.0, 0, '0'), Text(10.0, 0, '10'), Text(20.0, 0, '20'), Text(30.0, 0, '30'), Text(40.0, 0, '40')])

(array([4.5, 5. , 5.5, 6. , 6.5, 7. , 7.5, 8. , 8.5]), [Text(0, 4.5, '4.5'), Text(0, 5.0, '5.0'), Text(0, 5.5, '5.5'), Text(0, 6.0, '6.0'), Text(0, 6.5, '6.5'), Text(0, 7.0, '7.0'), Text(0, 7.5, '7.5'), Text(0, 8.0, '8.0'), Text(0, 8.5, '8.5')])

Linearity

• If \(f_1(x)\) and \(f_2(x)\) have Fourier series,
• Then for any constants \(a, b\),
• \(a f_1(x) + b f_2(x)\) has a Fourier series,
• With coefficients scaled accordingly.

Time Shifting

• If \(f(x)\) has Fourier coefficients \(a_n, b_n\),
• Then \(f(x - x_0)\) has coefficients:
• \(a_n \cos(n\omega x_0) + b_n \sin(n\omega x_0)\),
• And \(b_n \cos(n\omega x_0) - a_n \sin(n\omega x_0)\).

Frequency Scaling

• If \(g(x) = f(cx)\),
• Then the period scales by \(c\),
• The fundamental frequency changes to \(c\omega\),
• Fourier coefficients adjust accordingly.

** Differentiation Property**

• If \(f(x)\) is differentiable,
• Then \(f'(x)\) has Fourier series,
• With coefficients scaled as \(n a_n, n b_n\),
• Higher frequencies get amplified.

Integration Property

• If \(f(x)\) has a Fourier series,
• Then \(\int f(x) dx\) has a Fourier series,
• With coefficients scaled as \(\frac{a_n}{n}, \frac{b_n}{n}\),
• Lower frequencies get emphasized.

Parseval’s Theorem

• The total signal energy is conserved,
• Energy in time domain equals energy in frequency domain,
• Given by:
• \(\sum (a_n^2 + b_n^2) = \frac{1}{T} \int |f(x)|^2 dx\).

Convolution Property

• Convolution in time domain,
• Is multiplication in Fourier series coefficients,
• If \(f_1\) and \(f_2\) are convoluted,
• Their Fourier coefficients multiply component-wise.

Discrete Time Fourier Series

• Represents periodic discrete signals using harmonics.
• Extends Fourier series to discrete-time domain.
• Fundamental in digital signal processing.
• Basis for the Discrete Fourier Transform (DFT).

Mathematical Expression

• A periodic sequence \(x[n]\) can be expressed as:
• \[x[n] = \sum_{k=0}^{N-1} C_k e^{j(2\pi k n / N)}\].
• The coefficients \(C_k\) are computed as:
• \(C_k = \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-j(2\pi k n / N)}\).

Periodicity and Symmetry

• The coefficients \(C_k\) repeat every \(N\).
• Ensures correct reconstruction of signals.
• Explains frequency domain representation.
• Basis for spectral analysis.

Key Properties

• Linearity: Superposition holds.
• Time Shift: Causes phase shift in coefficients.
• Parseval’s Theorem: Energy conservation.
• Convolution: Time convolution → Frequency multiplication.

Frequency Domain Interpretation

• \(C_k\) represents discrete frequency content.
• The spectrum consists of \(N\) harmonics.
• Resolution improves with larger \(N\).
• Essential for analyzing periodic discrete signals.

Comparison with Continuous Case

• DTFS applies to discrete periodic signals.
• Continuous Fourier series applies to continuous functions.
• Both represent signals as sums of sinusoids.
• DTFS is used in digital communications and audio processing.

Example of th DTFS

DTFS Coefficients:

C[0] = 2.0000+0.0000j
C[1] = -0.6036-0.6036j
C[2] = 0.0000+0.0000j
C[3] = 0.1036-0.1036j
C[4] = 0.0000+0.0000j
C[5] = 0.1036+0.1036j
C[6] = 0.0000+0.0000j
C[7] = -0.6036+0.6036j

Example 02

Conceptual Foundation

• Fourier Series represents periodic signals in terms of sinusoids.
• As period \(T \to \infty\), the signal becomes aperiodic.
• The Fourier Transform generalizes Fourier Series to aperiodic signals.
• Transforms signals from time to frequency domain.

Mathematical Transition

• Fourier Series of a periodic signal:
• \[f(x) = \sum_{n=-\infty}^{\infty} C_n e^{j(2\pi n x / T)}\].
• As \(T \to \infty\), frequency spacing \(\frac{1}{T}\) → differential.
• Leads to the Fourier Transform:
• \[F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-j\omega x} dx\].

Frequency Spectrum Interpretation

• Fourier Series: discrete frequency spectrum.
• Fourier Transform: continuous frequency spectrum.
• Coefficients \(C_n\) become the function \(F(\omega)\).
• Allows analysis of arbitrary signals in frequency domain.

Inverse Fourier Transform

• Recovers time-domain signal from \(F(\omega)\).
• Defined as:
• \[f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega x} d\omega\].
• Ensures complete information preservation.
• Basis for signal reconstruction in DSP.

Energy and Parseval’s Theorem

• Energy conservation in time and frequency domains.
• Parseval’s theorem states:
• \[\int |f(x)|^2 dx = \frac{1}{2\pi} \int |F(\omega)|^2 d\omega\]
• Ensures no energy loss between domains.